food and nutrition


Grains

Question: What mass of sand grains would have a total surface area equal to the surface area of a cube 0.8 m on an edge? Grains of fine California beach sand are approximately spheres with an average radius of 50 μm and are made of silicon dioxide, which has a density of 2.3 × 103 kg/m3. What mass of sand grains would have a total surface area (the total area of all the individual spheres) equal to the surface area of a cube 0.8 m on an edge?

Answer: I've had a few beers tonight but thought I'd try this as it seemed fun! Ok, lets say the radius of the sand is r so the surface area of one sand is 4πr^2 so the surface area of n grains is A = n4πr^2 If ρ is the density of of silicon dioxide then the mass of on grain is (4/3)πr^3*ρ so the mass of n grains is m = n(4/3)πr^3*ρ Now A must equal the surface of the cube of side d So A = 6d^2 Therefore 6d^2 = n4πr^2 so rearrange for n n = 3d^2 / (2πr^2) Now put this n into the equation for m and your away m = n(4/3)πr^3*ρ m = 2d^2*r*ρ m = 2(0.8^2)*50*10^(-6)*2.3*10^3 = 0.1472 kg So m = 147 g


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